广义多重融合弦网模型的弱Hopf对称性与管代数
Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
AI总结：
本文研究了广义多重融合弦网模型的弱Hopf对称性，构造了其弱Hopf管代数，并利用幺正多重融合范畴刻画了带隙边界与畴壁相的对称性及管代数结构。
摘要：
我们研究了弦网基态和格点哈密顿量的多重融合推广，并深入探讨了其相关的弱Hopf对称性。对于多重融合弦网，规范对称性表现为一般的弱Hopf代数，导致可约的真空弦标；荷对称性作为规范对称性的量子偶，构成一个连通的弱Hopf代数。这表明相关的拓扑相仍保持由幺正模张量范畴（UMTC）刻画。体荷对称性也可由弱Hopf管代数刻画。我们给出了弱Hopf管代数结构的显式构造，并充分讨论了其性质。我们广泛讨论了带隙边界和畴壁模型，这些$1d$相由幺正多重融合范畴（UMFCs）刻画。我们深入探讨了这些$1d$相的规范与荷对称性，以及边界和畴壁管代数的构造。此外，我们阐明畴壁管代数可视为两个边界管代数的交叉积。作为模型的应用，我们阐明了如何将缺陷弦网解释为受限的多重融合弦网。
英文摘要：
We investigate the multifusion generalization of string-net ground states and lattice Hamiltonians, delving into its associated weak Hopf symmetry. For the multifusion string-net, the gauge symmetry manifests as a general weak Hopf algebra, leading to a reducible vacuum string label; the charge symmetry, serving as a quantum double of gauge symmetry, constitutes a connected weak Hopf algebra. This implies that the associated topological phase retains its characterization by a unitary modular tensor category (UMTC). The bulk charge symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit construction of the weak Hopf tube algebra structure and thoroughly discuss its properties. The gapped boundary and domain wall models are extensively discussed, with these $1d$ phases characterized by unitary multifusion categories (UMFCs). We delve into the gauge and charge symmetries of these $1d$ phases, as well as the construction of the boundary and domain wall tube algebras. Additionally, we illustrate that the domain wall tube algebra can be regarded as a cross product of two boundary tube algebras. As an application of our model, we elucidate how to interpret the defective string-net as a restricted multifusion string-net.
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